Rhombohedron

Rhombohedron
Type	prism
Faces	6 rhombi
Edges	12
Vertices	8
Symmetry group	Ci , [2+,2+], (×), order 2
Properties	convex, equilateral, zonohedron, parallelohedron

It is a special case of a parallelepiped where all edges are the same length. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A cube is a special case of a rhombohedron with all sides square.

In general a rhombohedron can have up to three types of rhombic faces in congruent opposite pairs, C_i symmetry, order 2.

Four points forming non-adjacent vertices of a rhombohedron necessarily form the four vertices of an orthocentric tetrahedron, and all orthocentric tetrahedra can be formed in this way.

The rhombohedral lattice system has rhombohedral cells, with 6 congruent rhombic faces forming a trigonal trapezohedron:

 

Form	Cube	Trigonal trapezohedron	Right rhombic prism	Oblique rhombic prism
Angle constraints	${\displaystyle \alpha =\beta =\gamma =90^{\circ }}$	${\displaystyle \alpha =\beta =\gamma }$	${\displaystyle \alpha =\beta =90^{\circ }}$	${\displaystyle \alpha =\beta }$
Symmetry	Oh order 48	D3d order 12	D2h order 8	C2h order 4
Faces	6 squares	6 congruent rhombi	2 rhombi, 4 squares	6 rhombi

• Cube: with O_h symmetry, order 48. All faces are squares.
• Trigonal trapezohedron (also called isohedral rhombohedron): with D_3d symmetry, order 12. All non-obtuse internal angles of the faces are equal (all faces are congruent rhombi). This can be seen by stretching a cube on its body-diagonal axis. For example, a regular octahedron with two regular tetrahedra attached on opposite faces constructs a 60 degree trigonal trapezohedron.
• Right rhombic prism: with D_2h symmetry, order 8. It is constructed by two rhombi and four squares. This can be seen by stretching a cube on its face-diagonal axis. For example, two right prisms with regular triangular bases attached together makes a 60 degree right rhombic prism.
• Oblique rhombic prism: with C_2h symmetry, order 4. It has only one plane of symmetry, through four vertices, and six rhombic faces.

For a unit (i.e.: with side length 1) isohedral rhombohedron, with rhombic acute angle ${\displaystyle \theta ~}$ , with one vertex at the origin (0, 0, 0), and with one edge lying along the x-axis, the three generating vectors are

e₁ : ${\displaystyle {\biggl (}1,0,0{\biggr )},}$ 

e₂ : ${\displaystyle {\biggl (}\cos \theta ,\sin \theta ,0{\biggr )},}$ 

e₃ : ${\displaystyle {\biggl (}\cos \theta ,{\cos \theta -\cos ^{2}\theta \over \sin \theta },{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }{\biggr )}.}$ 

The other coordinates can be obtained from vector addition of the 3 direction vectors: e₁ + e₂ , e₁ + e₃ , e₂ + e₃ , and e₁ + e₂ + e₃ .

The volume ${\displaystyle V}$  of an isohedral rhombohedron, in terms of its side length ${\displaystyle a}$  and its rhombic acute angle ${\displaystyle \theta ~}$ , is a simplification of the volume of a parallelepiped, and is given by

${\displaystyle V=a^{3}(1-\cos \theta ){\sqrt {1+2\cos \theta }}=a^{3}{\sqrt {(1-\cos \theta )^{2}(1+2\cos \theta )}}=a^{3}{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }}~.}$ 

We can express the volume ${\displaystyle V}$  another way :

${\displaystyle V=2{\sqrt {3}}~a^{3}\sin ^{2}\left({\frac {\theta }{2}}\right){\sqrt {1-{\frac {4}{3}}\sin ^{2}\left({\frac {\theta }{2}}\right)}}~.}$ 

As the area of the (rhombic) base is given by ${\displaystyle a^{2}\sin \theta ~}$ , and as the height of a rhombohedron is given by its volume divided by the area of its base, the height ${\displaystyle h}$  of an isohedral rhombohedron in terms of its side length ${\displaystyle a}$  and its rhombic acute angle ${\displaystyle \theta }$  is given by

${\displaystyle h=a~{(1-\cos \theta ){\sqrt {1+2\cos \theta }} \over \sin \theta }=a~{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }~.}$ 

Note:

${\displaystyle h=a~z}$ ₃ , where ${\displaystyle z}$ ₃ is the third coordinate of e₃ .

The body diagonal between the acute-angled vertices is the longest. By rotational symmetry about that diagonal, the other three body diagonals, between the three pairs of opposite obtuse-angled vertices, are all the same length.

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